Stability of solutions to abstract evolution equations with delay

An equation (u) over dot = A(t)u + B(t)F(t, u(t - tau)), u(t) = v(t), -tau <= t <= 0, is considered, where A(t) and B(t) are linear operators in a Hilbert space H, (u) over dot = du/dt, F : H -> H is a non-linear operator, and tau > 0 is a constant. Under some assumptions on A(t), B(t) and F(t, u) sufficient conditions are given for the solution u(t) to exist globally, i.e., for all t >= 0, to be globally bounded, and to tend to zero at a specified rate as t -> infinity. (C) 2012 Elsevier Inc. All rights reserved.